Exam P Master Formula Sheet Overview

A set of flashcards based on key formulas and distributions covered in the Exam P Master Formula Sheet, focusing on discrete and continuous probability distributions, their properties, and essential statistical concepts.

Binomial Distribution

X ∼ Bin(n, p) PMF: P(X = k) = (n choose k) p^k q^(n−k) where q = 1 − p.

Poisson Distribution

X ∼ Pois(λ) PMF: P(X = k) = (e^−λ λ^k)/k! where λ is the average rate.

Geometric Distribution

X ∼ Geom(p) represents the number of trials until the first success; PMF: P(X = k) = q^(k−1)p.

Negative Binomial Distribution

X ∼ NB(r,p) indicates the number of trials until the r-th success; PMF: P(X = k) = (k-1 choose r-1) p^r q^(k−r).

Hypergeometric Distribution

X ∼ Hyper(N,K,n) represents n draws without replacement from a population of size N with K successes; PMF uses combinatorial counts.

Bernoulli Distribution

X ∼ Bern(p) represents a single trial with two outcomes; PMF: P(X=1)=p and P(X=0)=q.

Discrete Uniform Distribution

X ∼ DU(a,b) represents a uniform distribution across n = b − a + 1 outcomes, with PMF = 1/n.

Expectation

E[X] is the mean of the random variable. For a linear transformation, E[aX + b] = aE[X] + b.

Variance

Var(X) measures the spread of a distribution. Var(X) = E[X²] − (E[X])².

Memoryless Property

A distribution is memoryless if P(X > s + t | X > s) = P(X > t). Holds for Geometric and Exponential distributions.

Moment Generating Function (MGF)

The MGF of a random variable X, MX(t), can be used to find all moments E[X^n] via derivatives.

Standard Normal Distribution

N(0, 1) with Z = (X − µ)/σ used for standardization of any normal variable.

Exponential Distribution

X ∼ Exp(λ) models time until the next event, with PDF: f(x) = λe^−λx.

Normal Distribution

X ∼ N(µ, σ²) represents a continuous distribution with bell-shaped curve, characterized by mean (µ) and variance (σ²).

Gamma Distribution

X ∼ Gamma(α, λ) generalizes exponential; PDF contains a shape parameter α.

Law of Total Expectation

E[X] = E[E[X | Y]] accounts for variability from conditioning on another variable.

Covariance

Cov(X,Y) = E[XY] - E[X]E[Y] measures how two random variables vary together.

CLT (Central Limit Theorem)

As sample size increases, the sampling distribution of the sample mean will approach a normal distribution.